Ineffable cardinal

In the mathematics of transfinite numbers, an ineffable cardinal is a certain kind of large cardinal number, introduced by Jensen & Kunen (1969). In the following definitions, ${\displaystyle \kappa }$ will always be a regular uncountable cardinal number.

A cardinal number ${\displaystyle \kappa }$ is called almost ineffable if for every ${\displaystyle f:\kappa \to {\mathcal {P}}(\kappa )}$ (where ${\displaystyle {\mathcal {P}}(\kappa )}$ is the powerset of ${\displaystyle \kappa }$) with the property that ${\displaystyle f(\delta )}$ is a subset of ${\displaystyle \delta }$ for all ordinals ${\displaystyle \delta <\kappa }$, there is a subset ${\displaystyle S}$ of ${\displaystyle \kappa }$ having cardinality ${\displaystyle \kappa }$ and homogeneous for ${\displaystyle f,}$ in the sense that for any ${\displaystyle \delta _{1}<\delta _{2}}$ in ${\displaystyle S}$, ${\displaystyle f(\delta _{1})=f(\delta _{2})\cap \delta _{1}}$.

A cardinal number ${\displaystyle \kappa }$ is called ineffable if for every binary-valued function ${\displaystyle f:[\kappa ]^{2}\to \{0,1\}}$, there is a stationary subset of ${\displaystyle \kappa }$ on which ${\displaystyle f}$ is homogeneous: that is, either ${\displaystyle f}$ maps all unordered pairs of elements drawn from that subset to zero, or it maps all such unordered pairs to one. An equivalent formulation is that a cardinal ${\displaystyle \kappa }$ is ineffable if for every sequence ${\displaystyle \langle A_{\alpha }:\alpha \in \kappa \rangle }$ such that each ${\displaystyle A_{\alpha }\subseteq \alpha }$, there is ${\displaystyle A\subseteq \kappa }$ such that ${\displaystyle \{\alpha \in \kappa :A\cap \alpha =A_{\alpha }\}}$ is stationary in κ.

Another equivalent formulation is that a regular uncountable cardinal ${\displaystyle \kappa }$ is ineffable if for every set ${\displaystyle S}$ of cardinality ${\displaystyle \kappa }$ of subsets of ${\displaystyle \kappa }$, there is a normal (i.e. closed under diagonal intersection) non-trivial ${\displaystyle \kappa }$-complete filter ${\displaystyle {\mathcal {F}}}$ on ${\displaystyle \kappa }$ deciding ${\displaystyle S}$: that is, for any ${\displaystyle X\in S}$, either ${\displaystyle X\in {\mathcal {F}}}$ or ${\displaystyle \kappa \setminus X\in {\mathcal {F}}}$. This is similar to a characterization of weakly compact cardinals.

More generally, ${\displaystyle \kappa }$ is called ${\displaystyle n}$-ineffable (for a positive integer ${\displaystyle n}$) if for every ${\displaystyle f:[\kappa ]^{n}\to \{0,1\}}$ there is a stationary subset of ${\displaystyle \kappa }$ on which ${\displaystyle f}$ is homogeneous (takes the same value for all unordered ${\displaystyle n}$-tuples drawn from the subset). Thus, it is ineffable if and only if it is 2-ineffable. Ineffability is strictly weaker than 3-ineffability.^{p. 399}

A totally ineffable cardinal is a cardinal that is ${\displaystyle n}$-ineffable for every ${\displaystyle 2\leq n<\aleph _{0}}$. If ${\displaystyle \kappa }$ is ${\displaystyle (n+1)}$-ineffable, then the set of ${\displaystyle n}$-ineffable cardinals below ${\displaystyle \kappa }$ is a stationary subset of ${\displaystyle \kappa }$.

Every ${\displaystyle n}$-ineffable cardinal is ${\displaystyle n}$-almost ineffable (with set of ${\displaystyle n}$-almost ineffable below it stationary), and every ${\displaystyle n}$-almost ineffable is ${\displaystyle n}$-subtle (with set of ${\displaystyle n}$-subtle below it stationary). The least ${\displaystyle n}$-subtle cardinal is not even weakly compact (and unlike ineffable cardinals, the least ${\displaystyle n}$-almost ineffable is ${\displaystyle \Pi _{2}^{1}}$-describable), but ${\displaystyle (n-1)}$-ineffable cardinals are stationary below every ${\displaystyle n}$-subtle cardinal.

A cardinal κ is completely ineffable if there is a non-empty ${\displaystyle R\subseteq {\mathcal {P}}(\kappa )}$ such that
- every ${\displaystyle A\in R}$ is stationary
- for every ${\displaystyle A\in R}$ and ${\displaystyle f:[\kappa ]^{2}\to \{0,1\}}$, there is ${\displaystyle B\subseteq A}$ homogeneous for f with ${\displaystyle B\in R}$.